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Luca Di Cerbo : Positivity in K\ahler-Einstein theory and hyperbolic geometry

We characterize logarithmic pairs which admit K\"ahler-Einstein metrics with negative scalar curvature and small cone-edge singularities along a simple normal crossing divisor. We show that if such K\"ahler-Einstein metrics exist for all small cone-angles then they exist for all angles in a fixed range depending on the dimension only. Remarkably, the existence of such a uniform range can be used to derive many interesting results in hyperbolic geometry. We give effective bounds on the number of cusped complex hyperbolic manifolds with given upper bound on the volume. We estimate the number of ends of such manifolds in terms of their volume. Finally, we discuss the projective algebraicity of minimal compactifications (Siu-Yau) of finite volume complex hyperbolic manifolds.

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